1) Introduction to basic concepts of numerical analysis: Approximation, relative error, conditioning of a numerical problem, review of analysis concepts ( Lagrange midpoint theorem, Taylor formula in one and more dimensions, order of infinitesimal).
2) Floating point representation of real numbers: Machine accuracy, cancellation of significant digits, examples.
3) Polynomial interpolation: Existence and unicity of the interpolating polynomial. Bounds on the approximation error. Newton's form for the interpolating polynomial. Composite polynomial interpolation.
4) Methods for nonlinear equations: Bisection method: error estimate and convergence proof. Newton's method and its variants (chord, secant method). Fixed point method: sufficient conditions for convergence.
5) Finite difference approximation of derivatives: Forward, backward and centered finite differences. Finite difference approximation of second order derivatives.Richardson extrapolation.
6) Numerical integration methods: Basic quadrature rules: midpoint, trapezoidal and Simpson rule. Composite integration rules. Error estimates for simple and composite rules. Numerical computation of Fourier series coefficients (FFT).
7) Numerical methods for ordinary differential equations: Overview of basic existence and uniqueness theorems. Examples of simple numerical methods: forward Euler, Heun, second order Runge Kutta, leapfrog, backward Euler, Crank Nicolson, Adams-Bashforth, higher order Runge-Kutta. Convergence of one step methods. A-stability of numerical methods. Extension to ODE systems.
8) Numerical methods for linear systems: Methods for upper and lower triangular systems. Gaussian elimination and LU factorization. Pivoting. Special cases of gaussian elimination: tridiagonal systems. Singular value decomposition. Condition number of a matrix and error analysis of numerical methods for linear systems.
9) The Poisson equation and elliptic problems in one spatial dimension. Finite difference numerical methods for the approximation of elliptic problems in one spatial dimension.
10) The linear, 1d advection-diffusion equation: Existence and uniqueness of solutions, representation formulae. Properties of the solution: regularity, maximum principle. Boundary conditions.
11) Application of Fourier series to the numerical solution of linear PDEs by separation
12) Finite difference methods for the linear, 1d advection diffusion equation: Examples of basic methods. Consistency, convergence and stability. Analysis of truncation error: numerical diffusion.